a) The neatness of the hole is a moot point, if you hide the hole! As long as the hole is just big enough for the screw to go through, it doesn't matter whether it's really pretty, because you should cover it up with a metal washer, which will make it pretty. The washer will also spread the strain out over a larger area, so the head of the screw won't pop through the hole in the elastic when you put clothes into the bag. Something like this fender washer is perfect:
You can get them at Home Depot in the Fasteners aisle, in the little orange bags.

It's a good idea to put these washers on the INSIDE, too, so the locking nuts won't pull through the hole you made in the liner and Velcro. Having washers on both sides also lets you torque down the locking nuts so they don't vibrate loose over time. Just don't over tighten, or you'll never get them off when you need to swap out the elastic in a year or two.

I use a variety of devices to punch holes in elastic and Velcro. I most commonly use an X-Acto knife, but I also use a leather punch (the squeeze type that work like an office hole punch) when I want a really precise size. I have also used scissors; fold the elastic in half at the point where you want the hole, nip the fold with the tip of the scissors, and repeat in the opposite direction, creating two crossed cuts. Trim the excess, and you have a diamond-shaped hole.

b) What I meant was, your elastic currently has the cut ends pointing up, out of the bag. If you turn them so they point down, into the bag, then the strap itself will be on top, making the screws a little less visible. Even so, you should fold over the elastic before you put the screws through it, so the screws will be going through two or even three thicknesses of the elastic, which will be much stronger than one, and hold the weight of your clothes without ripping apart..